| L(s) = 1 | + 2.49·2-s − 0.614·3-s + 4.23·4-s − 3.00·5-s − 1.53·6-s − 3.07·7-s + 5.59·8-s − 2.62·9-s − 7.50·10-s + 4.26·11-s − 2.60·12-s − 13-s − 7.69·14-s + 1.84·15-s + 5.49·16-s − 5.89·17-s − 6.54·18-s − 6.59·19-s − 12.7·20-s + 1.89·21-s + 10.6·22-s − 2.54·23-s − 3.43·24-s + 4.02·25-s − 2.49·26-s + 3.45·27-s − 13.0·28-s + ⋯ |
| L(s) = 1 | + 1.76·2-s − 0.355·3-s + 2.11·4-s − 1.34·5-s − 0.627·6-s − 1.16·7-s + 1.97·8-s − 0.873·9-s − 2.37·10-s + 1.28·11-s − 0.752·12-s − 0.277·13-s − 2.05·14-s + 0.476·15-s + 1.37·16-s − 1.42·17-s − 1.54·18-s − 1.51·19-s − 2.84·20-s + 0.413·21-s + 2.26·22-s − 0.531·23-s − 0.702·24-s + 0.804·25-s − 0.489·26-s + 0.665·27-s − 2.46·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1339 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1339 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 + T \) |
| 103 | \( 1 + T \) |
| good | 2 | \( 1 - 2.49T + 2T^{2} \) |
| 3 | \( 1 + 0.614T + 3T^{2} \) |
| 5 | \( 1 + 3.00T + 5T^{2} \) |
| 7 | \( 1 + 3.07T + 7T^{2} \) |
| 11 | \( 1 - 4.26T + 11T^{2} \) |
| 17 | \( 1 + 5.89T + 17T^{2} \) |
| 19 | \( 1 + 6.59T + 19T^{2} \) |
| 23 | \( 1 + 2.54T + 23T^{2} \) |
| 29 | \( 1 - 8.32T + 29T^{2} \) |
| 31 | \( 1 + 7.12T + 31T^{2} \) |
| 37 | \( 1 - 9.12T + 37T^{2} \) |
| 41 | \( 1 + 9.69T + 41T^{2} \) |
| 43 | \( 1 - 1.36T + 43T^{2} \) |
| 47 | \( 1 - 5.12T + 47T^{2} \) |
| 53 | \( 1 + 8.95T + 53T^{2} \) |
| 59 | \( 1 - 6.98T + 59T^{2} \) |
| 61 | \( 1 - 8.09T + 61T^{2} \) |
| 67 | \( 1 - 8.50T + 67T^{2} \) |
| 71 | \( 1 + 12.8T + 71T^{2} \) |
| 73 | \( 1 + 9.20T + 73T^{2} \) |
| 79 | \( 1 + 12.6T + 79T^{2} \) |
| 83 | \( 1 - 10.9T + 83T^{2} \) |
| 89 | \( 1 - 9.04T + 89T^{2} \) |
| 97 | \( 1 - 5.67T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.091828602105276594935254961806, −8.310019904935906704900651632310, −6.99404534985594098899107047920, −6.52910877414402773678417591136, −5.96379449783184788794318590007, −4.65040082669128106838827803052, −4.09735584579576699211471058077, −3.38892895009606868664407645735, −2.39251683092717718251239306558, 0,
2.39251683092717718251239306558, 3.38892895009606868664407645735, 4.09735584579576699211471058077, 4.65040082669128106838827803052, 5.96379449783184788794318590007, 6.52910877414402773678417591136, 6.99404534985594098899107047920, 8.310019904935906704900651632310, 9.091828602105276594935254961806
